Skip to main content Accessibility help
×
Home

Characterizations of Besov-Type and Triebel–Lizorkin–Type Spaces via Averages on Balls

  • Ciqiang Zhuo (a1), Winfried Sickel (a2), Dachun Yang (a3) and Wen Yuan (a3)

Abstract

Let $\ell \in \mathbb{N}$ and $\alpha \in (0,2\ell )$ . In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence ${{\{f-{{B}_{\ell ,{{2}^{-k}}}}f\}}_{k}}$ consisting of the difference between $f$ and the ball average ${{B}_{\ell ,{{2}^{-k}}}}f$ . These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization of Morrey–Sobolev spaces and ${{\text{Q}}_{\alpha }}$ spaces with $\alpha \in (0,1)$ , which are of independent interest.

Copyright

References

Hide All
[1] Adams, D. R. and J. Xiao, Morrey spaces in harmonic analysis. Ark. Mat. 50(2012), 201230. http://dx.doi.Org/1 0.1007/s11512-010-0134-0
[2] Alabern, R., Mateu, J., and Verdera, J., A new characterization ofSobolev spaces on R”. Math. Ann. 354(2012), 589626. http://dx.doi.Org/1 0.1007/s00208-011 -0738-0
[3] Chang, D.-C., Liu, J., Yang, D., and Yuan, W., Littlewood-Paley characterizations of Hajlasz-Sobolev and Triebel-Lizorkin spaces via averages on balls. Potential Anal. (2016). http://dx.doi.Org/10.1007/s11118-016-9579-5
[4] Chiarenza, F. and Frasca, M., Morrey spaces and Hardy-Littlewood maximal function, Rend. Math. 7 (1987), 273279.
[5] Dafni, G. and Xiao, J., Some new tent spaces and duality theorems for fractional Carleson measures and Qa (R“). J. Funct. Anal. 208(2004), 377422. http://dx.doi.Org/10.1016/SOO22-1236(3)00181-2
[6] Dafni, G. and Xiao, J., The dyadic structure and atomic decomposition ofQ spaces in several real variables. Tohoku Math. J. (2) 57(2005), 119145. http://dx.doi.Org/!0.2748/tmj/111 3234836
[7] Dai, F., Gogatishvili, A., Yang, D., and Yuan, W., Characterizations ofSobolev spaces via averages on balls. Nonlinear Anal. 128(2015), 8699. http://dx.doi.Org/10.1016/j.na.2O15.07.024
[8] Dai, F., Gogatishvili, A., Yang, D., and Yuan, W., Characterizations ofBesov and Triebel-Lizorkin spaces via averages on balls. J. Math. Anal. Appl. 433(2016), 13501368. http://dx.doi.Org/10.1016/j.jmaa.2015.08.054
[9] Dai, F., Liu, J., Yang, D., and Yuan, W., Littlewood-Paley characterizations of fractional Sobolev spaces via averages on balls. arxiv:1 511.07598
[10] DeVore, R. A. and Lorentz, G. G., Constructive approximation. Grundlehren der Mathematischen Wissenschaften, 303, Springer-Verlag, Berlin, 1993. http://dx.doi.Org/10.1007/978-3-662-02888-9
[11] El Baraka, A., An embedding theorem for Campanato spaces. Electron. J. Differential Equations 66(2002), 117.
[12] El Baraka, A., Function spaces ofBMO and Campanato type. In: Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., 9, Southwest Texas State Univ., San Marcos, TX, 2002, pp. 109115.
[13] El Baraka, A., Littlewood-Paley characterization for Campanato spaces. J. Funct. Spaces Appl. 4(2006), 193220. http://dx.doi.Org/10.1155/2006/921520
[14] Essen, M., Janson, S., Peng, L., and Xiao, J., Q spaces of several real variables. Indiana Univ. Math. J. 49(2000), 575615. http://dx.doi.Org/10.1512/iumj.2000.49.1732
[15] Fefferman, C. and Stein, E. M., Some maximal inequalities. Amer. J. Math. 93(1971), 107115. http://dx.doi.Org/1 0.2307/2373450
[16] Frazier, M. and Jawerth, B., Decomposition ofBesov spaces. Indiana Univ. Math. J. 34(1985), 777799. http://dx.doi.Org/10.1512/iumj.1 985.34.34041
[17] Frazier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1990), 34170. http://dx.doi.Org/10.101 6/0022-1236(90)90137-A
[18] Frazier, M., Jawerth, B., and Weiss, G., Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics, 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. http://dx.doi.Org/10.1090/cbms/079
[19] Gogatishvili, A., Koskela, P., and Zhou, Y., Characterizations ofBesov and Triebel-Lizorkin spaces on metric measure spaces. Forum Math. 25(2013), 787819. doil0.1515/form.2011.135
[20] Grafakos, L., Classical Fourier analysis. Third ed., Graduate Texts in Mathematics, 249, Springer, New York, 2014. http://dx.doi.Org/10.1007/978-1-4939-1194-3
[21] Koskela, P., Yang, D., and Zhou, Y., Pointwise characterizations ofBesov and Triebel-Lizorkin spaces and quasiconformal mappings. Adv. Math. 226(2011), 35793621. http://dx.doi.Org/1 0.101 6/j.aim.2O10.10.020
[22] Kozono, H. and Yamazaki, M., Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Differential Equations 19(1994), 9591014. http://dx.doi.Org/10.1080/03605309408821042
[23] Sawano, Y. and Tanaka, H., Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z. 257(2007), 871905. http://dx.doi.Org/10.1007/s00209-007-0150-3
[24] Sawano, Y., Yang, D., and Yuan, W., New applications of Besov-type and Triebel-Lizorkin-type spaces. J. Math. Anal. Appl. 363(2010), 7385. http://dx.doi.Org/10.1016/j.jmaa.2OO9.O8.OO2
[25] Sickel, W., Smoothness spaces related to Morrey spaces-a survey. I. Eurasian Math. J. 3(2012), 110149.
[26] Stein, E. M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.
[27] Tang, L. and Xu, J., Some properties of Morrey type Besov-Triebel spaces. Math. Nachr. 278(2005), 904917. http://dx.doi.Org/10.1002/mana.200310281
[28] Triebel, H., Theory of function spaces., Monographs in Mathematics, 78, Birkhauser Verlag, Basel, 1983. http://dx.doi.Org/10.1007/978-3-0346-0416-1
[29] Triebel, H., Hybrid function spaces, heat and Navier-Stokes equations. EMS Tracts in Mathematics, 24, European Mathematical Society (EMS), Zurich, 2014.
[30] Triebel, H., Tempered homogeneous function spaces. EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zurich, 2015. http://dx.doi.Org/10.4171/1 55
[31] Yang, D. and Yuan, W., A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J. Funct. Anal. 255(2008), 27602809. http://dx.doi.Org/10.1016/j.jfa.2008.09.005
[32] Yang, D. and Yuan, W., New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces. Math. Z. 265(2010), 451480. http://dx.doi.Org/10.1007/s00209-009-0524-9
[33] Yang, D. and Yuan, W., Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces. Appl. Anal. 92(2013), 549561. http://dx.doi.Org/! 0.1080/00036811.2011.629610
[34] Yang, D. and Yuan, W., Pointwise characterizations ofBesov and Triebel-Lizorkin spaces in terms of averages on balls. Trans. Amer. Math. Soc. (2016). http://dx.doi.Org/10.1090/tran/6871
[35] Yang, D., Yuan, W., and Zhou, Y., A new characterization of Triebel-Lizorkin spaces on W. Publ. Mat. 57(2013), 5782. http://dx.doi.Org/10.5565/PUBLMAT_57113_O2
[36] Yuan, W., Sickel, W., and Yang, D., Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, 2005, Springer-Verlag, Berlin, 2010. http://dx.doi.Org/10.1007/978-3-642-14606-0
[37] Yuan, W., Sickel, W., and Yang, D., On the coincidence of certain approaches to smoothness spaces related to Morrey spaces. Math. Nachr. 286(2013), 15711584.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed